More Applications of the Pumping Lemma

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More Applications ofthe Pumping Lemma
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The Pumping Lemma

• Given a infinite regular language

• there exists an integer

• for any string with length

• we can write

• with and

• such that

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Non-regular languages
Regular languages
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Theorem
The language
is not regular
Proof
Use the Pumping Lemma
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Assume for contradiction that is a regular
language
Since is infinite we can apply the Pumping
Lemma
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Let be the integer in the Pumping Lemma
Pick a string such that
and
length
We pick
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Write
From the Pumping Lemma
it must be that length
Thus
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From the Pumping Lemma
Thus
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From the Pumping Lemma
Thus
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BUT
CONTRADICTION!!!
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Our assumption that is a regular language is not
true
Therefore
Conclusion
is not a regular language
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Non-regular languages
Regular languages
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Theorem
The language
is not regular
Proof
Use the Pumping Lemma
14
Assume for contradiction that is a regular
language
Since is infinite we can apply the Pumping
Lemma
15
Let be the integer in the Pumping Lemma
Pick a string such that
and
length
We pick
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Write
From the Pumping Lemma
it must be that length
Thus
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From the Pumping Lemma
Thus
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From the Pumping Lemma
Thus
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BUT
CONTRADICTION!!!
20
Our assumption that is a regular language is not
true
Therefore
Conclusion
is not a regular language
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Non-regular languages
Regular languages
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Theorem
The language
is not regular
Proof
Use the Pumping Lemma
23
Assume for contradiction that is a regular
language
Since is infinite we can apply the Pumping
Lemma
24
Let be the integer in the Pumping Lemma
Pick a string such that
length
We pick
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Write
From the Pumping Lemma
it must be that length
Thus
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From the Pumping Lemma
Thus
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From the Pumping Lemma
Thus
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Since
There must exist such that
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However
for
for any
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BUT
CONTRADICTION!!!
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Our assumption that is a regular language is not
true
Therefore
Conclusion
is not a regular language